CN112904881A - Design method for dynamic gain scheduling controller of hypersonic aircraft - Google Patents

Design method for dynamic gain scheduling controller of hypersonic aircraft Download PDF

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CN112904881A
CN112904881A CN202110068853.9A CN202110068853A CN112904881A CN 112904881 A CN112904881 A CN 112904881A CN 202110068853 A CN202110068853 A CN 202110068853A CN 112904881 A CN112904881 A CN 112904881A
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gain
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王茜
林庆霞
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Hangzhou Dianzi University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/08—Control of attitude, i.e. control of roll, pitch, or yaw
    • G05D1/0808—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft
    • G05D1/0816—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability
    • G05D1/0833—Control of attitude, i.e. control of roll, pitch, or yaw specially adapted for aircraft to ensure stability using limited authority control

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Abstract

The invention discloses a design method of a hypersonic aircraft dynamic gain scheduling controller, and provides a design method of a continuous dynamic gain scheduling controller, aiming at the problem that when the existing hypersonic aircraft control system designs a controller on the basis of a hypersonic aircraft dynamic model, the influence on the stability of the system due to the change of system speed, dynamic pressure and other factors in the flight process cannot be avoided. Aiming at the hypersonic aircraft system with actuator saturation, the invention designs a continuous dynamic gain scheduling controller based on low-gain feedback control, thereby avoiding the occurrence of actuator saturation and improving the dynamic performance of the hypersonic aircraft system.

Description

Design method for dynamic gain scheduling controller of hypersonic aircraft
Technical Field
The invention belongs to the field of modern aircraft control, and designs a continuous dynamic gain scheduling controller aiming at improving the dynamic performance of a hypersonic aircraft. By designing the continuous dynamic gain scheduling controller with the actuator saturated switching system, the control target of improving the dynamic performance of the hypersonic aircraft system is realized, and the control method is suitable for controlling the hypersonic aircraft.
Background
With the development of science and technology, the aerospace technical field is also continuously improved, and the technical breakthrough of the hypersonic flight vehicle has very important significance for the international strategic pattern, military comparison and civil aviation. Even have profound effects on the comprehensive national forces of the country. Therefore, the dynamic performance of the hypersonic flight vehicle in flight is of research value.
The hypersonic aircraft integrates a plurality of leading-edge technologies of aerospace and aviation, so that the hypersonic aircraft has the characteristics of complex aerodynamic characteristics, high model nonlinearity degree, large flying height and speed span, complex flying environment and the like. Dynamic pressure, model non-linearity, and sudden height and velocity spans all contribute to its stability during its flight. This also makes stability control of hypersonic aircraft more and more difficult.
At present, in the existing hypersonic aircraft control system, a controller is designed on the basis of a hypersonic aircraft dynamic model, and the influence on the stability of the system caused by the change of system speed, dynamic pressure and other factors in the flight process cannot be avoided. Therefore, the control method is designed to avoid the influence of the change of factors such as system speed, dynamic pressure and the like on the stability of the system, and has important significance in improving the dynamic performance of the hypersonic aircraft system.
Disclosure of Invention
Aiming at the defects of the existing control method, the traditional control method is difficult to quickly reach stability due to the complexity of a hypersonic aircraft model. We propose a continuous dynamic gain scheduling controller to improve the dynamic characteristics of aircraft systems.
The invention provides a design method of a hypersonic aircraft dynamic gain scheduling controller, which is characterized in that a flight envelope of an aircraft is partitioned according to the speed and the dynamic pressure of the aircraft, so that the influence of the change of factors such as the speed and the dynamic pressure of a system on the stability of the system in the flight process of the system is avoided. The control method and the control device realize the control target of improving the dynamic performance of the hypersonic aircraft.
The method comprises the following specific steps:
step 1, establishing a state space model of a hypersonic aircraft
Defining a system state space model
Figure BDA0002905264400000021
Wherein X ═ V h α θ Q Φ Ψ]ΤRepresenting a state vector, where V represents aircraft speed, h represents aircraft altitude, α represents aircraft angle of attack, θ represents aircraft pitch,q represents the aircraft pitch angle rate, Φ represents the aircraft engine fuel ratio,
Figure BDA0002905264400000022
u＝[Φ δe δc]Τto control an input vector, where δeRepresenting the aircraft's elevator angle, deltacRepresenting the aircraft forward wing deflection angle.
Figure BDA00029052644000000216
Is a constant matrix; σ (t) represents a switching signal, from the set
Figure BDA0002905264400000024
A medium value, wherein M is an integer greater than 1. And (4) partitioning the flight envelope of the aircraft according to the speed and the dynamic pressure of the aircraft into M subsystems. sat (. cndot.) is a saturation function having the following form
sat(u)＝[sat(u1) sat(u2) … sat(um)]T
And is
Figure BDA0002905264400000025
I[1,m]The expression set {1,2,3.., m }, m ≧ 1, and superscript T denotes the transpose of the matrix. Hypothesis system
Figure BDA0002905264400000026
Is stable and matrix
Figure BDA0002905264400000027
All lie in the closed left half-plane, so that there is a non-singular matrix T, with
Figure BDA0002905264400000028
Figure BDA0002905264400000029
Figure BDA00029052644000000210
Wherein the content of the first and second substances,
Figure BDA00029052644000000211
is a constant matrix with eigenvalues in the left half-plane of the open,
Figure BDA00029052644000000212
for a constant matrix with characteristic values lying on the imaginary axis, ns+na7. T is a non-singular transformation matrix and is not unique. Since the characteristic value is located in the left half plane of the open, the stability of the system is not affected, and therefore, when considering the stability of the system, only the condition that the characteristic values are all on the virtual axis needs to be studied, that is, the following system is considered:
Figure BDA00029052644000000213
wherein the content of the first and second substances,
Figure BDA00029052644000000214
the control gain of the system is represented as a constant matrix.
Step 2, designing an ellipsoid set
The following two sets were designed:
Figure BDA00029052644000000215
Figure BDA0002905264400000031
wherein ξi(t) > 0 is a time-varying low gain parameter.
Figure BDA0002905264400000032
Is a symmetric positive definite matrix. i denotes running to the ith subsystem,
Figure BDA0002905264400000033
| | | represents the 2 norm of the vector or matrix.
Order to
Figure BDA0002905264400000034
Then, when
Figure BDA0002905264400000035
Time of flight
Figure BDA0002905264400000036
Can obtain
Figure BDA0002905264400000037
Then for any
Figure BDA0002905264400000038
sat(u(t))＝u(t)。
Step 3, designing a dynamic gain scheduling controller and average residence time
Designing a dynamic gain scheduling controller
Figure BDA0002905264400000039
Wherein, BiIndicating the controller gain, ξi(t) > 0 is a time-varying low gain parameter of the form
Figure BDA00029052644000000310
Wherein the content of the first and second substances,
Figure BDA00029052644000000311
ξi(0)＜λ＜2ξi(0) wherein, λ is a normal number, niTo representDimension of ith subsystem, ξi(0) Indicating the initial value of the i-th subsystem low-gain parameter. Thetaci＝θci(ξi(0) 1) is a normal number and can be calculated as follows
Figure BDA00029052644000000312
Wherein U (ξ)i(t)) can be solved by the following parametric Lyapunov equation
Figure BDA00029052644000000313
Time-varying low-gain parameter of the above-described form for any given initial value ξi(0) > 0 will converge to a bounded value, which can be calculated by a low gain parameter expression. Average residence time is satisfied
Figure BDA00029052644000000314
Where μ is a constant greater than 1. P (xi)i(t)) > 0 is a symmetric positive definite matrix that can be solved by the following parametric Riccati equation:
Ai TP(ξi(t))+P(ξi(t))Ai-P(ξi(t))BiBi TP(ξi(t))＝-ξi(t)P(ξi(t))
step 4, stability analysis
Substituting the designed controller (1) into a hypersonic aircraft state space model to obtain a closed loop system
Figure BDA0002905264400000041
According to the Lyapunov stability theorem, a Lyapunov function is selected
Vi(x,t)＝η(ξi(t))xTP(ξi(t))x
To make a closed loop systemIs stable only by
Figure BDA0002905264400000042
To make the patient feel
Figure BDA0002905264400000043
Then only need to
Figure BDA0002905264400000044
In which ξi(0)＜λ＜2ξi(0). Then, we can get the following equation
Figure BDA0002905264400000045
Wherein
Figure BDA0002905264400000046
Then we can get
Figure BDA0002905264400000047
To make it possible to
Figure BDA0002905264400000048
The following differential equation can be obtained
Figure BDA0002905264400000049
Solving the differential equation can obtain the expression of the time-varying low-gain parameter in step 3. Then, can obtain
Figure BDA00029052644000000410
I.e. the closed loop system is stable if the average residence time in step 3 is met.
The invention has the characteristics and beneficial effects that:
the invention provides a design method of a continuous dynamic gain scheduling controller aiming at the defects of the existing hypersonic aircraft control method. Aiming at the hypersonic aircraft system with actuator saturation, the invention designs a continuous dynamic gain scheduling controller based on low-gain feedback control, thereby avoiding the occurrence of actuator saturation and improving the dynamic performance of the hypersonic aircraft system.
Detailed Description
A design method for a dynamic gain scheduling controller of a hypersonic aircraft specifically comprises the following steps:
step 1, establishing a state space model of a hypersonic aircraft
Defining a system state space model
Figure BDA00029052644000000411
Wherein X ═ V h α θ Q Φ Ψ]ΤRepresenting a state vector, wherein V represents aircraft speed, h represents aircraft altitude, α represents aircraft angle of attack, θ represents aircraft pitch, Q represents aircraft pitch rate, Φ represents aircraft engine fuel ratio,
Figure BDA0002905264400000051
to control the input vector, δ represents the aircraft's elevator yaw angle, and δ represents the aircraft's forward wing yaw angle.
Figure BDA00029052644000000516
Is a constant matrix. σ (t) represents a switching signal, from the set
Figure BDA0002905264400000053
A medium value, wherein M is an integer greater than 1. And (4) partitioning the flight envelope of the aircraft according to the speed and the dynamic pressure of the aircraft into M subsystems. sat (. cndot.) is a saturation function having the following form
sat(u)＝[sat(u1) sat(u2) … sat(um)]T
And is
Figure BDA0002905264400000054
I[1,m]The expression set {1,2,3.., m }, m ≧ 1, and superscript T denotes the transpose of the matrix. Hypothesis system
Figure BDA0002905264400000055
Is stable and matrix
Figure BDA0002905264400000056
All lie in the closed left half-plane, so that there is a non-singular matrix T, with
Figure BDA0002905264400000057
Figure BDA0002905264400000058
Figure BDA0002905264400000059
Wherein the content of the first and second substances,
Figure BDA00029052644000000510
is a constant matrix with eigenvalues in the left half-plane of the open,
Figure BDA00029052644000000511
for a constant matrix with characteristic values lying on the imaginary axis, ns+na7. T is a non-singular transformation matrix and is not unique. Since the eigenvalue is located in the left half-plane of the open, which does not affect the stability of the system, we only need to study the case where the eigenvalues are all on the imaginary axis when considering the stability of the system, i.e., consider the following system:
Figure BDA00029052644000000512
wherein the content of the first and second substances,
Figure BDA00029052644000000513
the control gain of the system is represented as a constant matrix.
Step 2, designing an ellipsoid set
The following two sets were designed:
Figure BDA00029052644000000514
Figure BDA00029052644000000515
wherein ξi(t) > 0 is a time-varying low gain parameter.
Figure BDA0002905264400000061
Is a symmetric positive definite matrix. i denotes running to the ith subsystem,
Figure BDA00029052644000000613
| | | represents the 2 norm of the vector or matrix.
Order to
Figure BDA0002905264400000062
Then, when
Figure BDA0002905264400000063
Time of flight
Figure BDA0002905264400000064
Can obtain
Figure BDA0002905264400000065
Then for any
Figure BDA0002905264400000066
sat(u(t))＝u(t)。
Step 3, designing a dynamic gain scheduling controller and average residence time
Designing a dynamic gain scheduling controller
Figure BDA0002905264400000067
Wherein, BiIndicating the controller gain, ξi(t) > 0 is a time-varying low gain parameter of the form
Figure BDA0002905264400000068
Wherein the content of the first and second substances,
Figure BDA0002905264400000069
ξi(0)＜λ＜2ξi(0) wherein, λ is a normal number, niDimension, ξ, representing the ith subsystemi(0) Indicating the initial value of the i-th subsystem low-gain parameter. Thetaci＝θci(ξi(0) 1) is a normal number and can be calculated as follows
Figure BDA00029052644000000610
Wherein U (ξ)i(t)) can be solved by the following parametric Lyapunov equation
Figure BDA00029052644000000611
Time-varying low-gain parameter of the above-described form for any given initial value ξi(0) > 0 will converge to a bounded value, which can be calculated by a low gain parameter expression. Average residence time is satisfied
Figure BDA00029052644000000612
Where μ is a constant greater than 1. P (xi)i(t)) > 0 is a symmetric positive definite matrix that can be solved by the following parametric Riccati equation:
Ai TP(ξi(t))+P(ξi(t))Ai-P(ξi(t))BiBi TP(ξi(t))＝-ξi(t)P(ξi(t))
step 4, stability analysis
Substituting the designed controller (1) into a hypersonic aircraft state space model to obtain a closed loop system
Figure BDA0002905264400000071
Defining a Lyapunov function according to the Lyapunov stability theorem
Vi(x,t)＝η(ξi(t))xTP(ξi(t))x
To stabilize the closed loop system, only one needs to be used
Figure BDA0002905264400000072
To make the patient feel
Figure BDA0002905264400000073
Then only need to
Figure BDA0002905264400000074
In which ξi(0)＜λ＜2ξi(0). Then, we can get the following equation
Figure BDA0002905264400000075
Wherein
Figure BDA0002905264400000076
Then we can get
Figure BDA0002905264400000077
To make it possible to
Figure BDA0002905264400000078
The following differential equation can be obtained
Figure BDA0002905264400000079
Solving the differential equation can obtain the expression of the time-varying low-gain parameter in step 3. Then, can obtain
Figure BDA00029052644000000710
I.e. the closed loop system is stable if the average residence time in step 3 is met.

Claims (1)

1. A design method for a dynamic gain scheduling controller of a hypersonic aircraft is characterized by comprising the following steps:
step 1, establishing a state space model of a hypersonic aircraft
Establishing a system state space model
Figure FDA0002905264390000011
Wherein X ═ V h α θ Q Φ Ψ]ΤRepresenting a state vector, wherein V represents aircraft speed, h represents aircraft altitude, α represents aircraft angle of attack, θ represents aircraft pitch, Q represents aircraft pitch rate, Φ represents aircraft engine fuel ratio,
Figure FDA0002905264390000012
u＝[Φ δe δc]Τto control an input vector, where δeRepresenting the aircraft's elevator angle, deltacRepresenting the aircraft front wing deflection angle;
Figure FDA0002905264390000013
is a constant matrix; σ (t) represents a switching signal, from the set
Figure FDA0002905264390000014
A medium value, wherein M is an integer greater than 1; dividing the flight envelope of the aircraft into M subsystems according to the speed and the dynamic pressure of the aircraft; sat (. cndot.) is a saturation function having the following form
sat(u)＝[sat(u1) sat(u2) … sat(um)]T
And is
Figure FDA0002905264390000015
I[1,m]Representing a set {1,2,3.., m }, wherein m is more than or equal to 1, and superscript T represents the transposition of the matrix; hypothesis system
Figure FDA0002905264390000016
Is stable and matrix
Figure FDA0002905264390000017
All lie in the closed left half-plane, so that there is a non-singular matrix T, with
Figure FDA0002905264390000018
Figure FDA0002905264390000019
Figure FDA00029052643900000110
Wherein the content of the first and second substances,
Figure FDA00029052643900000111
is a constant matrix with eigenvalues in the left half-plane of the open,
Figure FDA00029052643900000112
for a constant matrix with characteristic values lying on the imaginary axis, ns+na7; t is a non-singular transformation matrix and is not unique; because the characteristic value is located in the left half plane of the open, the stability of the system is not affected, and therefore, when considering the stability of the system, only the condition that the characteristic values are all on the virtual axis needs to be studied, that is, the following system is considered:
Figure FDA0002905264390000021
wherein the content of the first and second substances,
Figure FDA0002905264390000022
representing the control gain of the system as a constant matrix;
step 2, designing an ellipsoid set
The following two sets were designed:
Figure FDA00029052643900000215
Figure FDA0002905264390000023
wherein ξi(t) > 0 is a time-varying low-gain parameter;
Figure FDA0002905264390000024
is a symmetric positive definite matrix; i denotes running to the ith subsystem,
Figure FDA0002905264390000025
| | represents a 2 norm of a vector or matrix;
order to
Figure FDA0002905264390000026
Then, when
Figure FDA0002905264390000027
Time of flight
Figure FDA0002905264390000028
To obtain
Figure FDA0002905264390000029
Then for any
Figure FDA00029052643900000210
sat(u(t))＝u(t)；
Step 3, designing a dynamic gain scheduling controller and average residence time
Designing a dynamic gain scheduling controller
Figure FDA00029052643900000211
Wherein, BiIndicating the controller gain, ξi(t) > 0 is a time-varying low gain parameter of the form
Figure FDA00029052643900000212
Wherein the content of the first and second substances,
Figure FDA00029052643900000213
ξi(0)＜λ＜2ξi(0) wherein, λ is a normal number, niDimension, ξ, representing the ith subsystemi(0) Indicates the ith subsystem lowAn initial value of the gain parameter; thetaci＝θci(ξi(0) Equal to or greater than 1 is a normal number, and is calculated by the following form
Figure FDA00029052643900000214
Wherein U (ξ)i(t)) is solved by the following parametric Lyapunov equation
Figure FDA0002905264390000031
Time-varying low-gain parameter of the above-described form for any given initial value ξi(0) A convergence to a bounded value is achieved when the value is more than 0, and the bounded value is calculated by a low-gain parameter expression; average residence time is satisfied
Figure FDA0002905264390000032
Wherein μ is a constant greater than 1; p (xi)i(t)) > 0 is a symmetric positive definite matrix, solved by the parametric Riccati equation:
Ai TP(ξi(t))+P(ξi(t))Ai-P(ξi(t))BiBi TP(ξi(t))＝-ξi(t)P(ξi(t))
step 4, stability analysis
Substituting the designed dynamic gain scheduling controller (1) into a hypersonic aircraft state space model to obtain a closed loop system
Figure FDA0002905264390000033
According to the Lyapunov stability theorem, a Lyapunov function is selected
Vi(x,t)＝η(ξi(t))xTP(ξi(t))x
To stabilize the closed loop system, only one needs to be used
Figure FDA0002905264390000034
To make the patient feel
Figure FDA0002905264390000035
Then only need to
Figure FDA0002905264390000036
In which ξi(0)＜λ＜2ξi(0) (ii) a The following equation is obtained
Figure FDA0002905264390000037
Wherein
Figure FDA0002905264390000038
Namely obtain
Figure FDA0002905264390000039
To make it possible to
Figure FDA00029052643900000310
The following differential equation can be obtained
Figure FDA00029052643900000311
Solving the differential equation to obtain an expression of the time-varying low-gain parameter in the step 3; then obtain
Figure FDA00029052643900000312
I.e. the closed loop system is stable if the average residence time in step 3 is met.
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